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Generalizing a result given by Li, Miao and Wang in 2022, we study the spectrality of a class of infinite convolutions in $\mathbb{R}^d$ generated by sequences of nearly $d$-th power lattices. This allows us to easily construct spectral measures with and without compact supports in $\mathbb{R}^d$. According to a result on the relation between supports of infinite convolutions and sets of infinite sums, we systematically study the Hausdorff and packing dimensions of infinite sums of finite sets in $\mathbb{R}^d$. As an application, we give concrete formulae for the Hausdorff and packing dimensions of the supports of a class of spectral measures in $\mathbb{R}^d$ with the form of infinite convolutions generated by specific sequences of nearly $d$-th power lattices, and finally we deduce that there are spectral measures with and without compact supports of arbitrary Hausdorff and packing dimensions in $\mathbb{R}^d$.